Integrals!

\[ \large \int_0^1 \int_0^1 \cdots \int_0^1 \left \{1 \div \prod_{n=1}^k x_n \right \} \, dx_1 \; dx_2 \; \cdots dx_k = 1 - \sum_{n=0}^{k-1} \dfrac{\gamma_n}{n!} \]

Prove that the equation above holds true where $\gamma_n$ denotes the $n^\text{th}$ Stieltjes constant, $\displaystyle \gamma_n := \lim_{m\to\infty} \left [ \sum_{k=1}^m \dfrac{(\ln k)^n}k - \dfrac{(\ln m)^{n+1}}{n+1}\right ]$ .

Notation: $\{ \cdot \}$ denotes the fractional part function.

Clarification: There are $k$ integrals.

This is a part of the set Formidable Series and Integrals

#Calculus

Easy Math Editor

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$